Aims
- Use gaussian processes (GPs) to resample Type II(b) SNe light curves
- Estimate some morphological parameters as defined by Pessi et al. (2019)
- Assess the appropriateness of using GPs for this purpose
- Goodness-of-fit of curves
- Clustering of SNe by morphology parameters
Expecting a sharp divide between Type II and Type IIb SNe.
Type IIb SNe Light Curves
Type II SNe Light Curves
Gaussian Process
- mean function, \(\mu(t)\)
- covariance or kernel function, \(k(t_i, t_j)\)
\[\boldsymbol{Y} = \begin{bmatrix} Y_1 \\ \vdots \\ Y_n \end{bmatrix} \sim \mathcal{GP}(\boldsymbol{\mu}, \boldsymbol{\Sigma})\] where \(\boldsymbol{\mu} = \mu(t_i),\; \boldsymbol{\Sigma} = \mathrm{Cov}(Y_i, Y_j) = k(t_i, t_j)\quad i,j = 1, \dots, n\).
\[\textrm{Squared Exponential} \qquad k(\tau; \lambda) = A \exp\left\{-\frac{1}{2}\left( \frac{\tau}{\lambda}\right)^2\right\}\]
\[\textrm{Matern-3/2}\qquad k(\tau; \lambda) = \left(1 + \sqrt{3}\left(\frac{\tau}{\lambda}\right)\right) \exp\left\{-\sqrt{3}\left(\frac{\tau}{\lambda}\right) \right\}\]
Dataset
- Twenty-one “high quality” lightcurves from the Open Supernova Catalog accessible by API:
- evenly sampled
- well-studied explosion
- chosen by visual inspection (!)
Methodology
Fit a GP using different kernels (RBF, Matern-3/2)
Visually assess the goodness-of-fit
- mean function (peak, plateau, linear decay)
Estimate the morphology parameters
- \(t_\textrm{rise}\): time between explosion to maximum light
- \(\Delta m_{40-30}\): mag. difference between phase 30 and 40
- dm1: the earliest maximum of first derivative
- dm2: the earliest minimum of second derivative
Results
Matern 3/2 vs RBF
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Different GP implementations
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First and Second Derivatives
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NB: Matern-3/2 processes are only 1-time differentiable.
Clustering by \(t_\textrm{rise}\) and \(\Delta m_{40-30}\)
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Conclusions
- Kernel choice is crucial, especially length-scale.
- Adding kernels together can partially fit complex behaviours at different scales.
- SN light curves are perhaps better fitted using non-stationary kernels that allow varying smoothness.
- Be cautious of different software implementations of kernel turning resulting in different results.
- Results are heavily dependent on the density of sampling.
- Still reproduced the clustering by Pessi et al. (2019)
Statistical claims needing caveats
- GPs tend to overfit
- easy to say when the physics and behaviour are known.
- selection of curves and goodness-of-fit was judged by visual inspection without consideration of variances.
- Don’t use models outside the range of their training data
- Depends on context, e.g., what about forecasting?
- GP interpolations not suited to estimating dm1 and dm2
- Matern-\(\nu\) kernels are only differentiable (“smooth”) up to \(\nu-1\) derivative.
Things to try
- Non-stationary GP kernels
- Uneven or sparsely sampled SNe light curves
- Incorporate uncertainty of observations